### 3.333 $$\int \frac{\sqrt{3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^8} \, dx$$

Optimal. Leaf size=194 $\frac{246159769 \left (2 x^2-x+3\right )^{3/2}}{866843099136 (2 x+5)^3}+\frac{19414831 \left (2 x^2-x+3\right )^{3/2}}{4013162496 (2 x+5)^4}-\frac{1464037 \left (2 x^2-x+3\right )^{3/2}}{13934592 (2 x+5)^5}+\frac{948341 \left (2 x^2-x+3\right )^{3/2}}{1741824 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}-\frac{12568315 (17-22 x) \sqrt{2 x^2-x+3}}{23776267862016 (2 x+5)^2}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{285315214344192 \sqrt{2}}$

[Out]

(-12568315*(17 - 22*x)*Sqrt[3 - x + 2*x^2])/(23776267862016*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(4032*
(5 + 2*x)^7) + (948341*(3 - x + 2*x^2)^(3/2))/(1741824*(5 + 2*x)^6) - (1464037*(3 - x + 2*x^2)^(3/2))/(1393459
2*(5 + 2*x)^5) + (19414831*(3 - x + 2*x^2)^(3/2))/(4013162496*(5 + 2*x)^4) + (246159769*(3 - x + 2*x^2)^(3/2))
/(866843099136*(5 + 2*x)^3) - (289071245*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(2853152143441
92*Sqrt[2])

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Rubi [A]  time = 0.267503, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {1650, 834, 806, 720, 724, 206} $\frac{246159769 \left (2 x^2-x+3\right )^{3/2}}{866843099136 (2 x+5)^3}+\frac{19414831 \left (2 x^2-x+3\right )^{3/2}}{4013162496 (2 x+5)^4}-\frac{1464037 \left (2 x^2-x+3\right )^{3/2}}{13934592 (2 x+5)^5}+\frac{948341 \left (2 x^2-x+3\right )^{3/2}}{1741824 (2 x+5)^6}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}-\frac{12568315 (17-22 x) \sqrt{2 x^2-x+3}}{23776267862016 (2 x+5)^2}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{285315214344192 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^8,x]

[Out]

(-12568315*(17 - 22*x)*Sqrt[3 - x + 2*x^2])/(23776267862016*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(4032*
(5 + 2*x)^7) + (948341*(3 - x + 2*x^2)^(3/2))/(1741824*(5 + 2*x)^6) - (1464037*(3 - x + 2*x^2)^(3/2))/(1393459
2*(5 + 2*x)^5) + (19414831*(3 - x + 2*x^2)^(3/2))/(4013162496*(5 + 2*x)^4) + (246159769*(3 - x + 2*x^2)^(3/2))
/(866843099136*(5 + 2*x)^3) - (289071245*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(2853152143441
92*Sqrt[2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}-\frac{1}{504} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{69381}{16}-5594 x+3402 x^2-1260 x^3\right )}{(5+2 x)^7} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}+\frac{\int \frac{\sqrt{3-x+2 x^2} \left (\frac{10506615}{16}-\frac{2815905 x}{4}+272160 x^2\right )}{(5+2 x)^6} \, dx}{217728}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}-\frac{\int \frac{\left (\frac{231748695}{16}-\frac{32095935 x}{2}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^5} \, dx}{78382080}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{\int \frac{\left (-\frac{2340515655}{16}+\frac{873667395 x}{4}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^4} \, dx}{22574039040}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}+\frac{12568315 \int \frac{\sqrt{3-x+2 x^2}}{(5+2 x)^3} \, dx}{82556485632}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}+\frac{289071245 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{47552535724032}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}-\frac{289071245 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{23776267862016}\\ &=-\frac{12568315 (17-22 x) \sqrt{3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac{948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac{1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac{19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac{246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}-\frac{289071245 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{285315214344192 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.206567, size = 96, normalized size = 0.49 $\frac{24 \sqrt{2 x^2-x+3} \left (1574342277056 x^6+27976951397184 x^5+4982916071952 x^4+41058010262368 x^3+14716683780036 x^2+590492177460 x-20465234808721\right )-2023498715 \sqrt{2} (2 x+5)^7 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{3994413000818688 (2 x+5)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^8,x]

[Out]

(24*Sqrt[3 - x + 2*x^2]*(-20465234808721 + 590492177460*x + 14716683780036*x^2 + 41058010262368*x^3 + 49829160
71952*x^4 + 27976951397184*x^5 + 1574342277056*x^6) - 2023498715*Sqrt[2]*(5 + 2*x)^7*ArcTanh[(17 - 22*x)/(12*S
qrt[6 - 2*x + 4*x^2])])/(3994413000818688*(5 + 2*x)^7)

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Maple [A]  time = 0.078, size = 216, normalized size = 1.1 \begin{align*}{\frac{948341}{111476736} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}-{\frac{138251465}{855945643032576} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{-138251465+553005860\,x}{1711891286065152}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{12568315}{23776267862016} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{289071245\,\sqrt{2}}{570630428688384}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }+{\frac{289071245}{1711891286065152}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{19414831}{64210599936} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}}+{\frac{246159769}{6934744793088} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{3667}{516096} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-7}}-{\frac{1464037}{445906944} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^8,x)

[Out]

948341/111476736/(x+5/2)^6*(2*(x+5/2)^2-11*x-19/2)^(3/2)-138251465/855945643032576/(x+5/2)*(2*(x+5/2)^2-11*x-1
9/2)^(3/2)+138251465/1711891286065152*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(1/2)-12568315/23776267862016/(x+5/2)^2
*(2*(x+5/2)^2-11*x-19/2)^(3/2)-289071245/570630428688384*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2
-11*x-19/2)^(1/2))+289071245/1711891286065152*(2*(x+5/2)^2-11*x-19/2)^(1/2)+19414831/64210599936/(x+5/2)^4*(2*
(x+5/2)^2-11*x-19/2)^(3/2)+246159769/6934744793088/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(3/2)-3667/516096/(x+5/2)
^7*(2*(x+5/2)^2-11*x-19/2)^(3/2)-1464037/445906944/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(3/2)

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Maxima [A]  time = 1.54707, size = 406, normalized size = 2.09 \begin{align*} \frac{289071245}{570630428688384} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) + \frac{12568315}{11888133931008} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4032 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac{948341 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1741824 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac{1464037 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{13934592 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{19414831 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4013162496 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{246159769 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{866843099136 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{12568315 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{5944066965504 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{138251465 \, \sqrt{2 \, x^{2} - x + 3}}{23776267862016 \,{\left (2 \, x + 5\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^8,x, algorithm="maxima")

[Out]

289071245/570630428688384*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 12568
315/11888133931008*sqrt(2*x^2 - x + 3) - 3667/4032*(2*x^2 - x + 3)^(3/2)/(128*x^7 + 2240*x^6 + 16800*x^5 + 700
00*x^4 + 175000*x^3 + 262500*x^2 + 218750*x + 78125) + 948341/1741824*(2*x^2 - x + 3)^(3/2)/(64*x^6 + 960*x^5
+ 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625) - 1464037/13934592*(2*x^2 - x + 3)^(3/2)/(32*x^5 + 400*x
^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) + 19414831/4013162496*(2*x^2 - x + 3)^(3/2)/(16*x^4 + 160*x^3 + 600*
x^2 + 1000*x + 625) + 246159769/866843099136*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 12568315/5
944066965504*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) - 138251465/23776267862016*sqrt(2*x^2 - x + 3)/(2*x + 5
)

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Fricas [A]  time = 1.43963, size = 649, normalized size = 3.35 \begin{align*} \frac{2023498715 \, \sqrt{2}{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \,{\left (1574342277056 \, x^{6} + 27976951397184 \, x^{5} + 4982916071952 \, x^{4} + 41058010262368 \, x^{3} + 14716683780036 \, x^{2} + 590492177460 \, x - 20465234808721\right )} \sqrt{2 \, x^{2} - x + 3}}{7988826001637376 \,{\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^8,x, algorithm="fricas")

[Out]

1/7988826001637376*(2023498715*sqrt(2)*(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x^3 + 262500*x^2 +
218750*x + 78125)*log(-(24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x
+ 25)) + 48*(1574342277056*x^6 + 27976951397184*x^5 + 4982916071952*x^4 + 41058010262368*x^3 + 14716683780036*
x^2 + 590492177460*x - 20465234808721)*sqrt(2*x^2 - x + 3))/(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 1750
00*x^3 + 262500*x^2 + 218750*x + 78125)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{8}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**4-x**3+3*x**2+x+2)*(2*x**2-x+3)**(1/2)/(5+2*x)**8,x)

[Out]

Integral(sqrt(2*x**2 - x + 3)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**8, x)

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Giac [B]  time = 1.34148, size = 616, normalized size = 3.18 \begin{align*} -\frac{289071245}{570630428688384} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{289071245}{570630428688384} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{\sqrt{2}{\left (129503917760 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{13} - 3320259746027840 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{12} - 23966708071916736 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} - 186055342532355520 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} - 274256644494948976 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 796135370176031760 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 2531523139171005408 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 4610393811900786336 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 7997126854300052364 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 30842713619423538868 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 21873571601855032556 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 16204706960604668100 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 3196254593191113265 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 536799032216117911\right )}}{332867750068224 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^8,x, algorithm="giac")

[Out]

-289071245/570630428688384*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 289071245/570630
428688384*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/332867750068224*sqrt(2)*(129
503917760*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^13 - 3320259746027840*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^12
- 23966708071916736*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 - 186055342532355520*(sqrt(2)*x - sqrt(2*x^2
- x + 3))^10 - 274256644494948976*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 796135370176031760*(sqrt(2)*x
- sqrt(2*x^2 - x + 3))^8 + 2531523139171005408*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 4610393811900786
336*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6 - 7997126854300052364*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 30
842713619423538868*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 - 21873571601855032556*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2
- x + 3))^3 + 16204706960604668100*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 3196254593191113265*sqrt(2)*(sqrt(2)*
x - sqrt(2*x^2 - x + 3)) + 536799032216117911)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x
- sqrt(2*x^2 - x + 3)) - 11)^7